The source is capable of supplying some number of amps, at some voltage, depending on how it is built.
Total Resistance
Consider a 1.5v battery being the total energy source and miles of thin copper wire. Lets say the wire equals 9Ω total resistance, and the battery has 1Ω of internal resistance, so the total resistance is 10Ω:
- The total Amperes it can supply to a shorted load (0Ω): 1.5v/10Ω = 0.150A
- The total Amperes it can supply to a medium load (10Ω): 1.5v/20Ω = 0.075A
- The total Amperes it can supply to a light load (1kΩ): 1.5v/1.01kΩ = 1.485mA
- The total Amperes it can supply to your dry fingers (100,0010Ω): 14.9985µA
Now lets change things around. Instead of one battery, we are going to use twenty of them, all wired in parallel. And instead of miles of thin wire, we're going to use short copper bus bars. Twenty batteries in parallel is still 1.5v, but the internal resistance (1Ω each) are also in parallel:
$$ R_P = \frac{1}{\frac{1}{R1}+\frac{1}{R2}+\frac{1}{Rn}...}$$
This makes the internal resistance equivalent to one battery with 0.05Ω. Lets say the bus bars are also 0.05Ω, for a total resistance of 0.1Ω. What happens now?
- The total Amperes it can supply to your dry fingers (100,000.1Ω): 14.999985µA
- The total Amperes it can supply to a light load (1kΩ): 1.5v/1000.1Ω = 1.49985mA
- The total Amperes it can supply to a medium load (10Ω): 1.5v/10.1Ω = 0.1485A
- The total Amperes it can supply to a shorted load (0Ω): 1.5v/0.1Ω = 15A!
If the delicate thing connected to this battery had 15A dumped into it, it would be destroyed.
Household (Mains) Power
Mains power works in a similar way, but is more dangerous mainly because the voltage is much higher (120v instead of 1.5v, or 80 times more voltage!)
What limits this current is both the load resistance and the source resistance. The total resistance is the key.
Now mains power has an intrinsically low resistance. Thick copper wires connect the utility service to the house, and relatively thick wire connects from here to the outlets. So something must be present to prevent current from getting too high, and that's exactly what a fuse and/or circuit breaker does.
If you have an outlet rated for 120vAC/15A and attempt to plug two hair dryers into it (30A load), a fuse or circuit breaker will trip. This may be inconvenient, but it's much better than the wiring in the walls catching fire. I'll let you in on a little secret though. Fuses and circuit breakers are not instantaneous. If you could somehow switch both of those hair dryers on at exactly the same time, you'd see 30A flowing for a very small amount of time. And if you shorted the outlet, at least 100A briefly... that much current is dangerous, and causes wires to melt and vaporize, which is highly unpleasant at best. Research ArcFlash for more info.
Skin Resistance
Now contact with fingers is a whole other story. Just like the "variable load" used above ranges from 0Ω to 100kΩ, so can your skin. Dry skin with a small contact area is relatively resistant to electricity. But wet skin and contaminants (sweat) greatly reduces the contact resistance, perhaps as low as 1kΩ. This net effect is a combination of factors:
- Dryer the skin - higher the resistance.
- Smaller the contact area - higher the resistance.
- Lower the voltage - less damage it can do.
Now this is only for the skin. Underneath the skin (inside the body) is very wet and so has a low resistance, perhaps a few Ohms. So if the voltage is high enough to break through the skin and reach inside, then LOTS of current can flow. This is why high voltage is so dangerous around humans - skin offers little protection.
You can however, touch a 12v car battery's terminals without as much as a sensation, even though that battery can deliver 1000A for a short time. Your skin protects you from that. But the skin cannot protect you from a high voltage source, such as an 18,400v power line. It only takes 10mA (0.01A) to be a risk of life.
See? Amps, Ohms, Volts - they are all related, and why it is called Ohm's Law. One is meaningless without the others.